Computing marginal likelihoods from a single MCMC output

In this article, we propose new Monte Carlo methods for computing a single marginal likelihood or several marginal likelihoods for the purpose of Bayesian model comparisons. The methods are motivated by Bayesian variable selection, in which the marginal likelihoods for all subset variable models are required to compute. The proposed estimates use only a single Markov chain Monte Carlo (MCMC) output from the joint posterior distribution and it does not require the specific structure or the form of the MCMC sampling algorithm that is used to generate the MCMC sample to be known. The theoretical properties of the proposed method are examined in detail. The applicability and usefulness of the proposed method are demonstrated via ordinal data probit regression models. A real dataset involving ordinal outcomes is used to further illustrate the proposed methodology.